Universal series induced by approximate identities and some relevant applications

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Journal of Approximation Theory 163 (2011) 1783–1797 www.elsevier.com/locate/jat

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Universal series induced by approximate identities and some relevant applications Vassili Nestoridis a , Sebastian Schmutzhard b,∗ , Vangelis Stefanopoulos c a Department of Mathematics, University of Athens, 15784 Panepistimiopolis, Greece b Faculty of Mathematics, University of Vienna, 1090 Vienna, Austria c Department of Mathematics, University of the Aegean, 83200 Karlovasi, Samos, Greece

Received 11 September 2010; received in revised form 21 March 2011; accepted 2 June 2011 Available online 12 June 2011 Communicated by Hans G Feichtinger

Abstract ∑ We prove the existence of series an ψn , whose coefficients (an ) are in ∩ p>1 ℓ p and whose terms (ψn ) are translates by rational vectors in Rd of a family of approximations to the identity, having the property that the partial sums are dense in various spaces of functions such as Wiener’s algebra W (C0 , ℓ1 ), Cb (Rd ), C0 (Rd ), L p (Rd ), for every p ∈ [1, ∞), and the space of measurable functions. Applying this theory to particular situations, we establish approximations by such series to solutions of the heat and Laplace equations as well as to probability density functions. c 2011 Elsevier Inc. Open access under CC BY-NC-ND license. ⃝ Keywords: Universal series; Wiener algebra; Approximation to the identity; Gauss kernel; Poisson kernel; Normal distribution; Heat equation; Laplace’s equation

1. Introduction Starting from any random variable and taking appropriate averages, we end up in the limit with a normal distribution. Conversely, starting from the normal distribution is it possible to ∗ Corresponding author.

E-mail addresses: [email protected] (V. Nestoridis), [email protected] (S. Schmutzhard), [email protected] (V. Stefanopoulos). c 2011 Elsevier Inc. Open access under CC BY-NC-ND license. 0021-9045 ⃝ doi:10.1016/j.jat.2011.06.001

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reach, by some limiting process, any random variable? The answer is in the affirmative, because 1 any probability ∑n density can be approximated in the L norm by a convex combination of normal densities k=1 ak φ(· − m k , bk ) [1,8]. It turns out that this approximation result can be put in the framework of universal series, strengthened significantly, and generalized. More precisely, in the present paper we prove the following results (see Theorem 3.1). Let {φϵ }ϵ>0 be an approximation to the identity on Rd , and let {ξn }∞ n=1 be an enumeration is an enumeration of all φ (· − ξ ), k, l ∈ N, then there exists a sequence of Qd . If {ψn }∞ l 1/k n=1 ∑∞ p (an )n≥1 in ∩ p>1 ℓ such that the partial sums of the series k=1 ak ψk are dense in W (C0 , ℓ1 ), the so-called Wiener algebra. Moreover, thanks to the density properties of W (C0 , ℓ1 ) to several classical spaces, we also show the following (see Theorem 3.2). ∑ The partial sums of the series ∞ k=1 ak ψk are also dense, with respect to their natural topologies, in L p (Rd ) for every p ∈ [1, ∞), the space of continuous functions vanishing at infinity C0 (Rd ), C0 ∩ L 1 (Rd ), the space of continuous and bounded functions Cb (Rd ), and the space of measurable functions L 0 (Rd ). Similar results can be obtained if we replace ∩ p>1 ℓ p by ℓq , 1 < q < ∞, c0 , or RN the space of all real sequences. We show, however, that the space ∩ p>1 ℓ p cannot be replaced by ℓ1 . The proof is based on the fact that, if (ψϵ )ϵ>0 is an approximation to the identity and f is a function in W (C0 , ℓ1 ) on Rd , then ψϵ ∗ f approximates f , as ϵ → 0 in various ways, and that each convolution ψϵ ∗ f , being an integral, can be approximated by a Riemann sum. Thus some linear combinations of translates of ψϵ approximate f . This fact, according to a recently developed abstract theory of universal series [9,2], implies the generic existence of such series, and in particular implies the above-mentioned approximation results. The same argument extends naturally to obtaining universal series for solutions to the heat or Laplace equations. Here, what we deal with is that every such solution fits together with initial or boundary data through a convolution with an appropriate approximation of the identity: the Gauss kernel for the parabolic case and the Poisson kernel for the elliptic one. The new element here is that we do not approximate the initial or boundary state but rather the solution itself. The series obtained are valuable in the context of partial differential equations (PDEs) since the derivatives of all orders of the approximating partial sums converge to the corresponding derivatives of the solution. The above approximation results and in particular those related to probability and partial differential equations are new contributions to the theory of universal series and lead outside the initial range of applications of Baire’s category theorem, as described in the excellent surveys on universal series [7,6]. This paper is organized as follows. Section 2 contains mainly background material on universal series and Wiener’s algebra W (C0 , ℓ1 ). In Section 3, we prove the main approximation results of the paper. In Section 4, we state the results of Section 3 in terms of probability densities. In Section 5, extending the approximation result of Section 3, we prove the existence of series whose partial sums converge uniformly on compact sets, along appropriate subsequences, to bounded solutions of the heat equation, in Rd . We also show that the derivatives of all orders of the partial sums converge to the derivatives of the corresponding solution. Finally, in Section 6, we obtain analogous approximation results to the ones obtained in Section 5 for solutions of the Dirichlet problem for Laplace’s equation in the upper half space of Rd+1 as well as in the unit ball of Rd . In all series, the coefficients belong to ∩ p>1 ℓ p .

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2. Preliminaries 2.1. Universal series Let X be a real vector space, and consider the metric space (X, ρ), where ρ is compatible with +, · and is invariant under translations. If (yn )n≥1 is a fixed sequence of ∑elements of X , by U we denote the class of sequences (an )n≥1 in R for which the partial sums nk=1 ak yk , n = 1, 2, . . . form a dense set in X . Observe that if (an )n≥1 ∈ U and x ∈ X then there exists a sequence of natural numbers (νn )n≥1 , which can be taken to satisfy νn < νn+1 for all n, such that   νn − lim ρ x, ak yk = 0. n→∞

k=1

Thus each (an )n≥1 ∈ U, whenever U ̸= ∅, generates an unrestricted universal series. Of special interest is the case where the coefficients of the universal series are required to belong to a particular subspace of the space of all real sequences, RN , where by N we denote the set of positive integers. These are the restricted universal series for which the following result is established in [9,2]. Let A be a vector subspace of the set of all sequences in R and let σ be a metric on A compatible with +, · that is invariant under translations. Setting e˜1 = (1, 0, 0, . . .), e˜2 = (0, 1, 0, . . .), . . ., we moreover assume that the following hold. (A1) (A, σ ) is complete. (A2) The projections A ∋ (an )n≥1 → am ∈ R are continuous for all m ∈ N. (A3) A ⊃ G = {(an )n≥1 : an ̸= 0∑ for only finitely many n’s}. (A4) If a˜ = (an )n≥1 ∈ A then σ ( nk=1 ak e˜k , a) ˜ → 0, as n → ∞. Then we have the following. Theorem 2.1 ([9,2]). Let X, (yn )n≥1 , U and A be as above, and let o˜ be the zero sequence of A. We set U A := U ∩ A. Then the following statements are equivalent. 1. U A ̸= ∅. 2. Given x ∈ X and ϵ > 0, there exist N ≥ 1 and b1 , b2 , . . . , b N in R such that     N N − − ρ x, bk yk < ϵ, σ o, ˜ bk e˜k < ϵ. k=1

k=1

3. U A is a dense G δ set in A, and U A ∪ {o} ˜ contains a dense vector subspace of A. In this paper, we deal with series whose coefficients are sequences belonging to the intersection of all ℓ p spaces for p > 1. Definition 2.1. Let p0 ∈ [1, ∞) be fixed and let pn := p0 + 1/n, n ∈ N. We define the metric space (A, σ ) by A := ∩ ℓ p , p> p0

σ (a, ˜ c) ˜ :=

∞ − ˜ pn 1 ‖a˜ − c‖ , n 1 + ‖a 2 ˜ − c‖ ˜ pn n=1

(2.1)

with a, ˜ c˜ ∈ A and ‖ · ‖ p being the usual ℓ p norm. We note that the definition above makes sense since ℓ p ⊂ ℓq , whenever p < q; hence ∩ p> p0 ℓ p = ∩n≥1 ℓ pn . The space (A, σ ) is a Fr´echet space, and one can check with no difficulty that the postulates (A1)–(A4) are satisfied.

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Lemma 2.1. Let (X, ρ) be a real vector space with translation-invariant metric ρ compatible with +, ·, and let (yn )n≥1 be a fixed sequence in X . Let U be the set of scalar sequences defining universal series. Let p0 ≥ 1 be fixed. The following statements are equivalent. (1) U ∩ ℓ p ̸= ∅, for every p > p0 . (2) U ∩ (∩ p> p0 ℓ p ) ̸= ∅. Proof. The direction (2) ⇒ (1) is obvious. We prove (1) ⇒ (2). Appealing to Theorem 2.1(2), it suffices to show that, given x ∈ X and ϵ > 0, there exist scalars c1 , . . . , c N so that the following approximations are accomplished:     N N − − ρ x, ck yk < ϵ, σ o, ˜ ck e˜k < ϵ. k=1

k=1

∑∞

Choose, first, n ∈ N so that k=n+1 < ϵ/2. Applying, next, (1) with p = pn , where, as before, pn = p0 + 1/n, and since U ∩ ℓ pn ̸= ∅, there exist c1 , . . . , c N so that     N N   − − ϵ ϵ   ck e˜k  < . ρ x, ck yk < , o˜ −   2 2 k=1 k=1 2−k

pn

∑N

Setting c˜ = k=1 ck e˜k ,and recalling that ‖ · ‖ p ≤ ‖ · ‖ pn , whenever p > pn , we compute, by the choice of n, n − ˜ pk 1 ‖c‖ ϵ σ (o, ˜ c) ˜ < + k 2 1 + ‖c‖ ˜ pk 2 k=1   1 ϵ ≤ 1 − n ‖c‖ ˜ pn + 2 2 < ϵ. Thus U ∩ (∩ p> p0 ℓ p ) ̸= ∅, and the proof is complete.



2.2. Wiener’s algebra W (C0 , ℓ1 ) As a kind of universal function space, we introduce the following. ∑ Definition 2.2. The space of all f ∈ C0 (Rd ) for which n∈Zd supx∈[0,1]d | f (x + n)| < ∞, endowed with the norm − ‖ f ‖W (C0 ,ℓ1 ) := sup | f (x + n)|, d n∈Zd x∈[0,1]

is a Banach space and is called Wiener’s algebra W (C0 , ℓ1 ). In what follows, whenever we write ‖ · ‖ we will always mean the W (C0 , ℓ1 ) norm. Simplifying notation, we set Q n := n + [0, 1]d for n ∈ Zd . Remark 2.1. One immediately sees that W (C0 , ℓ1 ) ⊂ L ∞ (Rd ) and that ‖ f ‖∞ ≤ ‖ f ‖, whenever f ∈ W (C0 , ℓ1 ). Moreover, W (C0 , ℓ1 ) is continuously and densely embedded in L p for 1 ≤ p ≤ ∞ (see [5, Chapter 6.1]). The proofs in this paper make use of several convenient properties of this space; in particular, the fact that it is densely and continuously embedded in several common function spaces, see for example [3] or [10], plays an important role.

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In the proof of the main outcome, we will need the following straightforward result. Lemma 2.2. Let f ∈ W (C0 , ℓ1 ). Then, for each x ∈ Rd , the translation Tx f := f (· − x) is in W (C0 , ℓ1 ), and ‖Tx f ‖ ≤ 2d ‖ f ‖.

(2.2)

Moreover, for a fixed f ∈ W (C0 , ℓ1 ), the map Rd ∋ x → Tx f (·) ∈ W (C0 , ℓ1 ) is continuous. Proof. Since, for each n ∈ Zd , supt∈Q n | f (t − x)| = supt∈Q n −x | f (t)|, assuming first that x ∈ Zd , we conclude that ‖Tx f ‖ = ‖ f ‖. Otherwise, each cube Q n − x, of unit edge, is contained in a cube with integral vertices with double-size edge. From this observation, (2.2) follows. Let x ∈ Rd and ϵ > 0 be given. Then −   Tx f − Ty f  = sup | f (t − x) − f (t − y)| n∈Zd t∈Q n



 −

=

+

|n|>2N x

−

sup | f (t − x) − f (t − y)|,

|n|≤2N x

t∈Q n

where N x ∈ N is arbitrary. The first sum may become arbitrarily small, less than ϵ/2 say, for the tail of a convergent series. Indeed, let Nϵ be such that ∑ the right choice of N x as2+d sup | f (t)| < ϵ/2 . Choose N x > Nϵ such that ∪|n|>2Nx Q n − z ⊂ ∪|n|>Nx Q n , t∈Q n |n|>Nϵ for z = x, y, where |x − y| < 1, say. Then − − − sup | f (t − x) − f (t − y)| ≤ sup | f (t − x)| + sup | f (t − y)| |n|>2N x t∈Q n

|n|>2N x t∈Q n

=

−

sup | f (t)| +

|n|>2N x t∈Q n −x

≤ 2d

−

|n|>2N x t∈Q n

−

sup | f (t)| + 2d

|n|>N x t∈Q n

sup | f (t)|

|n|>2N x t∈Q n −y

−

sup | f (t)|.

|n|>N x t∈Q n

The second sum may become less than ϵ/2, for y sufficiently close to x, by the uniform continuity of f .  Remark 2.2. This lemma shows that Wiener’s algebra is a Segal algebra with the translationinvariant and equivalent norm ‖ f ‖′ := sup ‖Tx f ‖. x

In fact, it is the first example of a Segal algebra. See also [10, Chapter 6.2]. 2.3. Approximations to the identity A family {Fϵ }ϵ>0 of integrable functions on Rd satisfying ∫ βϵ θ (i) Fϵ (x) dx = 1, (ii) |Fϵ (x)| ≤ , (ϵ + |x|)d+θ Rd

(2.3)

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where β and θ are positive constants independent of ϵ, is called an approximation to the identity, or approximate identity. The reason is explained by the fact that, if f : Rd → R is an integrable function and f ϵ := Fϵ ∗ f , that is ∫ ∫ f ϵ (x) := Fϵ (x − y) f (y) dy = Fϵ (y) f (x − y) dy, Rd

Rd

then f ϵ → f , as ϵ → 0, in various ways. In the context of Wiener’s algebra, we have the following. Lemma 2.3. Assume that the family {Fϵ }ϵ>0 satisfies conditions (2.3). Let f ∈ W (C0 , ℓ1 ) and f ϵ = Fϵ ∗ f . Then f ϵ ∈ W (C0 , ℓ1 ) and f ϵ → f , in the W (C0 , ℓ1 ) norm, as ϵ → 0. Proof. An easy calculation shows that ∫ Rd

βωd , θ

|Fϵ (x)| dx ≤

∫ |x|>δ

|Fϵ (x)| dx ≤

βωd θ



ϵ ϵ+δ

θ

,

(2.4)

where ωd is the area of the unit sphere in Rd . Therefore, if f ∈ W (C0 , ℓ1 ), then ∫  −    ‖ fϵ ‖ = sup  Fϵ (y) f (x − y) dy  Rd

n∈Zd x∈Q n

∫ ≤ Rd

|Fϵ (y)|

−

sup | f (x − y)| dy

n∈Zd x∈Q n

β ≤ 2d ‖ f ‖ ωd , θ via (2.2) and (2.4), and thus f ϵ ∈ W (C0 , ℓ1 ). Next, via the continuity result of Lemma 2.2, (2.2) and (2.4), we compute ∫  −    ‖ fϵ − f ‖ = sup  Fϵ (y)[ f (x − y) − f (x)] dy  n∈Zd x∈Q n

∫ ≤ Rd

|Fϵ (y)|

∫ = |y|<δ

≤

Rd

−

sup | f (x − y) − f (x)| dy

n∈Zd x∈Q n

|Fϵ (y)|‖Ty f − f ‖ dy +

βωd βωd ϵ+ θ θ



ϵ ϵ+δ

θ

∫ |y|≥δ

|Fϵ (y)|‖Ty f − f ‖ dy

(2d + 1)‖ f ‖,

for δ sufficiently small. Hence the result follows.



Remark 2.3. This result is shared by all Segal algebras. In [10], Proposition 6.2.4 shows that Segal algebras are ideals in L 1 , and the proof of Proposition 6.2.8 in the same book shows the approximation property. Moreover, Lemma 2.3 is valid for all homogeneous Banach spaces, if the functions Fϵ are generated by one function (see [4]), as is done below.

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The typical way of constructing approximate identities is that, given a function F ∈ L 1  satisfying F(x) dx = 1 and |F(x)| ≤ β(1 + |x|)−d−θ , one defines Fϵ (x) :=

1 x  F , ϵd ϵ

x ∈ Rd , ϵ > 0.

(2.5)

Then each Fϵ satisfies conditions (i) and (ii) of (2.3). 3. The main result We consider a family of continuous functions {φϵ }ϵ>0 satisfying (2.3), where β and θ are constants independent of ϵ. Note that, due to the assumed continuity and property (2.3)(ii), it is easy to show that φϵ ∈ W (C0 , ℓ1 ) for all ϵ > 0. Definition 3.1. We set {ψn }∞ n=1 to be an arbitrary but fixed enumeration of the countable collection of the functions {φk −1 (· − ξ ) : (k, ξ ) ∈ N × Qd }; that is, ψn (x) = φk −1 (x − ξ ),

for some k ∈ N and ξ ∈ Qd .

(3.1)

p Theorem 3.1. Let {ψn }∞ n=1 be as in (3.1). Then there exists a sequence (an )n≥1 in ∩ p>1 ℓ with 1 the property that, given a function f ∈ W (C0 , ℓ ), there exists an increasing sequence (νn )n≥1 in N such that   νn −    lim  (3.2) ak ψk − f  = 0. n→∞   k=1

Moreover, the set of such universal sequences is a dense G δ set in ∩ p>1 ℓ p , and it contains a dense vector subspace of ∩ p>1 ℓ p except the zero sequence. Proof. We prove the theorem by showing that condition (2) of Theorem 2.1 is fulfilled. For a given approximation error ϵ, we choose a k0 so that ‖ f − f ∗ φk −1 ‖ ≤ ϵ/3 for all k ≥ k0 (see Lemma 2.3). We will show that, due to the properties of Wiener’s algebra W (C0 , ℓ1 ), and in fact this would work for every Segal algebra, it is possible to approximate the convolution f ∗φk −1 by a Riemann sum, and furthermore to use only translates by rational vectors. First, we approximate the convolution by an integral over a rational cube K . ‖ f ∗ φk −1 − ( f χ K ) ∗ φk −1 ‖ = ‖( f χ K c ) ∗ φk −1 ‖ ∫  −   = sup  f (y)φk −1 (x − y) dy  Kc

n∈Zd x∈Q n

∫ ≤ Kc

−

| f (y)|

≤ 2d ‖φk −1 ‖

sup |φk −1 (x − y)| dy

n∈Zd x∈Q n

∫ Kc

| f (y)| dy,

via (2.2). For K large enough, the  right-hand side gets smaller than ϵ/3. Since we can interpret the convolution f χ K ∗φk −1 (x) = K f (y)φk −1 (x − y) dy as a Riemann integral, we can approximate it in the W (C0 , ℓ1 ) sense by Riemann sums, which are, in this situation, weighted sums of translates of φk −1 . It is enough to take the translations on a uniform, rational grid K g in K with

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‘gridsize’ η, i.e. K = ∪ y∈K g y + [0, η]d , where, for y1 , y2 ∈ K g with y1 ̸= y2 , the intersections {y1 + [0, η)d } ∩ {y2 + [0, η)d } are empty sets. Reshuffling sums and integrals gives     −   d ( f χ K ) ∗ φ −1 −  η f (y)φ (· − y) −1 k k     y∈K g    − ∫ −   [ f (y + t)φk −1 (x − y − t) − f (y)φk −1 (x − y)] dt  = sup  d  n∈Zd x∈Q n  y∈K g [0,η] ∫ − − sup | f (y + t)φk −1 (x − y − t) − f (y)φk −1 (x − y)| dt ≤ y∈K g

≤

[0,η]d n∈Zd x∈Q n

− ∫ y∈K g

− [0,η]d

 sup | f (y + t)φk −1 (x − y − t) − f (y)φk −1 (x − y − t)|

n∈Zd x∈Q n

 + | f (y)φk −1 (x − y − t) − f (y)φk −1 (x − y)| dt, from which, using the continuity result of Lemma 2.2, and the fact that f is uniformly continuous, by taking η small enough, leads us to − ∫ ≤ (| f (y + t) − f (y)|‖Ty+t φk −1 ‖ + | f (y)|‖Tt φk −1 − φk −1 ‖) dt y∈K g

[0,η]d

≤ 2d ‖φk −1 ‖ |K | sup ‖Tt f − f ‖∞ + |K |‖ f ‖∞ sup ‖Tt φk −1 − φk −1 ‖ t∈[0,η]d

t∈[0,η]d

ϵ . 3 So we have found that     −   d f − η f (y)φk −1 (· − y)   < ϵ,   y∈K g <

and therefore   N   −   cn ψn  < ϵ f −   n=1 for an N ∈ N and the right choice of the coefficients cn . Notice that there are fewer than 2d |K |/ηd coefficients in this sum, all smaller than or equal to ηd ‖ f ‖∞ . So the ℓ p -norm of the coefficients is p smaller than or equal to (ηdp ‖ f ‖∞ 2d |K |/ηd )1/ p = Cηd( p−1)/ p , which could be made arbitrary small by again choosing η sufficiently small. Lemma 2.1 finishes the proof.  ∑ Remark 3.1. We note at this point that there exists no universal series ak ψk with ℓ1 coefficients. Indeed, supposing that such a series exists, we are led via (2.4) to ∫ ∞ ∞ − β − |ak | |ψk (x)| dx ≤ ωd |ak | < ∞. θ Rd k=1 k=1 Thus the series converges almost everywhere on Rd to an L 1 (Rd ) function. This is, however, in contradiction with the fact that its partial sums are dense in W (C0 , ℓ1 ).

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1 Theorem 3.2. Let {ψn }∞ n=1 ⊂ W (C 0 , ℓ ) be as in (3.1). The universal sequence (an )n≥1 ∈ p ∩ p>1 ℓ provided by Theorem 3.1 also satisfies the following. (1) Given a function f ∈ L q (Rd ), with 1 ≤ q < ∞, there exists an increasing sequence (νn )n≥1 in N such that   νn   −   ak ψk  = 0. lim  f − n→∞   k=1 q

(2) Given a function f ∈ C0 (Rd ), that is, f is continuous and vanishes at infinity, there exists an increasing sequence (νn )n≥1 in N such that   νn   −   lim sup  f (x) − ak ψk (x) = 0. n→∞   x∈Rd k=1 (3) Given a function f ∈ C0 (Rd ) ∩ L 1 (Rd ), there exists an increasing sequence (νn )n≥1 in N such that     νn νn     − −     lim  f − a k ψk  +  f − ak ψk  = 0. n→∞     k=1 k=1 1

∞

(4) Given a function f ∈ Cb (Rd ), that is, f is continuous and bounded, there exists an increasing sequence (νn )n≥1 in N such that   νn   −   ak ψk (x) = 0 lim sup  f (x) − n→∞ x∈K   k=1 for every compact K ⊂ Rd . (5) Given a measurable function f , there exists an increasing sequence (νn )n≥1 in N such that νn − ak ψk → f, a.e. as n → ∞. k=1

(6) Let µ be a σ -finite Borel measure on Rd . Then, given a µ-measurable function f , there exists an increasing sequence (νn )n≥1 in N such that νn − ak ψk → f, µ—a.e. as n → ∞. k=1

Proof. The assertions in (1), (2), (3), and (4) are easily shown by the fact that W (C0 , ℓ1 ) is densely and continuously embedded in L q , 1 ≤ q < ∞, C0 , C0 ∩ L 1 and Cb with respect to the natural topologies, respectively. (5) A combination of Lusin’s theorem and Urysohn’s lemma renders a sequence of continuous functions with compact support (gk )k≥1 converging to f almost everywhere. (2) implies that for each k ∈ N there exists νk such that ‖gk − Sνk ‖∞ < 1/k. Then, the sequence of partial sums (Sνk )k≥1 converges to f almost everywhere. (6) The proof is similar to the one in (5).  Remark 3.2. In Theorems 3.1 and 3.2, we proved existence of universal series with coefficients in ∩ p>1 ℓ p . The same result holds in ℓq , q > 1, and c0 , since ∩ p>1 ℓ p ⊂ ℓq ⊂ c0 , as well as for the space of all sequences RN . In particular, the generic results contained in statement (3) of Theorem 2.1 hold in the ℓq , q > 1, c0 and RN settings.

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4. Universal series and probability densities In this section, we consider approximations of the identity generated, via (2.5), by probability densities. To simplify the presentation, we consider the one-dimensional case. Possible such densities are the Laplace √ density: f (x) = λ exp(−λ|x|)/2, λ > 0, the standard normal density: f (x) = exp(−x 2 /2)/ 2π , the Cauchy density: f (x) = θ/[π(x 2 +θ 2 )], θ > 0, or any other continuous density. Hence, we prove the existence of universal series whose terms are modified rational translates of these densities. Translating Theorem 3.2 in this setting, we have the following. Proposition 4.1. Let φ be a continuous probability density and let {rn }∞ n=1 be an arbitrary but fixed enumeration of the set N × Q. For rn = (k, ξ ), we set φrn (x) = kφ(k(x − ξ )), and we ∞ p consider ∑ the family {φrn }n=1 . Then, there exists a sequence (an )n≥1 in ∩ p>1 ℓ such that the series an φrn has the properties of the series of Theorems 3.1 and 3.2. Moreover, the set of such universal sequences (an )n≥1 is a dense G δ set in ∩ p>1 ℓ p , and it contains a dense vector subspace of ∩ p>1 ℓ p except the zero sequence. Remark 4.1. Notice that, if φ is the standard normal density, then φk −1 (x − ξ ) = φξ,k −2 (x),

(4.1)

the normal density with mean ξ and variance 1/k 2 . In analogy to statement (1) of Theorem 3.2 for q = 1, we recover the known result that any probability distribution is approximated in the L 1 norm by a finite linear combination of normal distributions. By Proposition 4.1 it follows, however, that this approximation is accomplished by other probability distributions as well and in every L q norm, q ≥ 1. 5. Universal series and the heat equation We consider the initial value problem for the heat equation x ∈ Rd , t > 0,

u t = ∆x u,

u(x, 0) = f (x),

(5.1a)

x ∈R , d

(5.1b)

where ∆x is the Laplacian and f is a continuous function. Let 2 /4t

G(x, t) := (4πt)−d/2 e−|x|

,

x ∈ Rd , t > 0,

(5.2)

and set G t (x) := G(x, t). Then the family {G t (x) : t > 0, x ∈ is an approximation to the identity. This is the Gauss or heat kernel. We recall that the function G(x − y, t) is a solution of (5.1a), for each y, and, if f is a continuous and bounded function, then the unique bounded solution of (5.1a) and (5.1b) is given by ∫ u(x, t) = G(x − y, t) f (y) dy, =: G(·, t) ∗1 f (x), Rd }

Rd

 and the initial condition is attained as f (x) = limt→0 Rd G(x − y, t) f (y) dy. Moreover, the map Cb (Rd ) ∋ f → G ∗1 f ∈ C ∞ (Rd × R+ ) is well defined and continuous if we regard the topologies of uniform convergence on compact sets on Cb (Rd ) and the uniform convergence on compact sets of arbitrary derivatives on

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C ∞ (Rd × R+ ). The latter topology is given by the semi-norms pn (u) :=

sup |x|≤n,1/n≤t≤n |α|≤n,r ≤n

|∂xα ∂tr u(x, t)|,

so the continuity inequalities pn (G ∗1 f ) ≤ Cn ‖ f ‖∞

(5.3)

are valid for the right choice of Cn > 0. Furthermore, C ∞ (Rd × R+ ) is a complete metric space with metric ρ(u, v) =

∞ − 1 pn (u − v) . n 1 + p (u − v) 2 n n=1

d Theorem 5.1. Let {ξn }∞ let us set n=1 be an enumeration of ∑all rational vectors in R and G n (x, t) := G(x − ξn , t). Then there exist series an G n with (an )n≥1 ∈ ∩ p>1 ℓ p , having the property that, given any bounded solution u of the heat equation with continuous and bounded initial data f , there exists an increasing sequence (νn )n≥1 in N such that

∂xα ∂tr

νn −

ak G k (x, t) → ∂xα ∂tr u(x, t),

as n → ∞

(5.4)

k=1

uniformly on compact sets of Rd × R+ , for every multi-index α with |α| ≥ 0 and for every nonnegative integer r . Moreover, the set of such universal sequences (an )n≥1 is a dense G δ set in ∩ p>1 ℓ p , and it contains a dense vector subspace of ∩ p>1 ℓ p except the zero sequence. Proof. We again show assertion (2) of Theorem 2.1 by inferring Lemma 2.1. Let u be the bounded solution of the heat equation with boundary data f ∈ W (C0 , ℓ1 ), or C0 , and ϵ > 0. For a compact set K ⊂ Rd × R+ there ∑ is an N ∈ N such that K ⊂ {|x| ≤ N , 1/N ≤ t ≤ N } on the one hand, and on the other k=N +1 2−k ≤ ϵ/3. Therefore, ρ(u, v) ≤ p N (u − v) + ϵ/3 holds for all v ∈ C ∞ (Rd × R+ ). Since u and its derivatives are uniformly continuous on {|x| ≤ 2N , 1/2N ≤ t ≤ 2N }, we can further choose 0 < τ0 < 1/2N such that ϵ p N (u(·, ·) − u(·, · − τ )) < 3 for all 0 < τ < τ0 . By the reasoning in the proof of Theorem 3.1, we can also take τ0 in a way that for 0 < τ < τ0 there are coefficients {bk }nk=1 ⊂ C and points {xk }nk=1 ⊂ Qd such that   n   − ϵ   bk G(· − xk , τ ) < f −   3C2N k=1 ∞

and  n −

1/ p |bk |

p

<ϵ

k=1

for a fixed p > 1 and C2N corresponding to the appropriate constant of inequality (5.3). Due to the continuity inequality in (5.3) mentioned above, and using the well-known property of the

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Gaussian kernel G(·, t) ∗ G(·, s) = G(·, s + t), we derive   n − ϵ bk G(· − xk , · + τ ) < . p2N u(·, ·) − 3 k=1 From this,  p N u(·, · − τ ) −

n −

 bk G(· − xk , ·) <

k=1

ϵ 3

follows, since τ < 1/2N . By the triangle inequality, we now establish     n n − − ϵ ρ u(·, ·), bk G(· − xk , ·) ≤ p N u(·, ·) − bk G(· − xk , ·) + 3 k=1 k=1 ≤ p N (u(·, ·) − u(·, · − τ ))   n − ϵ + p N u(·, · − τ ) − bk G(· − xk , ·) + 3 k=1 ≤ ϵ.

Again, by continuity and density, this result is also valid for f ∈ Cb . Lemma 2.1 finishes the proof.  ∑ Note 5.1. If a sequence of partial sums of the series an G n approximates the solution u, then the same sequence approximates∑ the initial condition f for t small, uniformly on compact sets of n Rd . Indeed, assuming that ρ(u, νk=1 an G n ) → 0 as n → ∞, then, for ϵ small, via Lemma 2.3, we have ‖ f − u(·, τ )‖∞ < ϵ/2. Hence, for 1/N < τ , we finally obtain   νn   −   sup  f (x) − an G n (x, τ )   |x|≤N k=1   νn   −   ≤ sup | f (x) − u(x, τ )| + sup u(x, τ ) − an G n (x, τ )  |x|≤N |x|≤N  k=1   νn − ≤ ‖ f − u(·, τ )‖∞ + p N u − an G n k=1

< ϵ, as long as p N (u −

∑ν n

k=1 an G n )

< ϵ/2.

6. Universal series and Laplace’s equation 6.1. The case of the upper half space Let H be the upper half space in Rd+1 , that is, H = {(x, y) ∈ Rd+1 : y > 0}, and consider the boundary value problem 1u = 0,

(x, y) ∈ H,

u(x, 0) = f (x),

x ∈ ∂ H,

(6.1a) (6.1b)

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where f is a bounded, continuous, real-valued function on Rd = ∂ H . The corresponding solution is given by ∫ u(x, y) = f (s)P(x − s, y) ds, x ∈ Rd , y > 0, (6.2) Rd

and the boundary value is realized as the limit of the integral as y → 0. The function P(x, y) in the integrand is the Poisson kernel for the upper half space, and it is defined by P(x, y) =

2 ωd+1

(|x|2

y , + y 2 )(d+1)/2

where ωd+1 is the surface area of the unit sphere in Rd+1 . P is a harmonic function, and, defining Py (x) := P(x, y), the family {Py (x) : y > 0, x ∈ Rd } is also an approximate identity, and hence a result that is analogous to the Theorem 5.1 result can be proved. Theorem 6.1. If {ξn }∞ of the rational vectors in Rd and Pn (x, y) := n=1 is an enumeration ∑ Py (x − ξn ), then there exist series an Pn with coefficients in ∩ p>1 ℓ p having the property that, given any function u harmonic on H , and with continuous and bounded boundary data, there exists an increasing sequence (νn )n≥1 in N such that ∂α

νn −

ak Pk (x, y) → ∂ α u(x, y),

as n → ∞

k=1

uniformly on compact sets of H , for every multi-index α, |α| ≥ 0. Moreover, the set of such universal sequences (an )n≥1 is a dense G δ set in ∩ p>1 ℓ p , and it contains a dense vector subspace of ∩ p>1 ℓ p except the zero sequence. Proof. Let u be a function harmonic on H and continuous on the closure   of H , and let K ⊂ H be compact. Since Rd P(x − s, y) ds = 1, for all y > 0, the integral u(s, 0)P(x − s, y) ds may become arbitrarily small outside a compact set containing K . Hence the integral in (6.2) is approximated uniformly on K by Riemann sums N −

u(ξk , 0)P(x − ξk , y)|K k |,

k=1

where K 1 , . . . , K N are cubes covering K . Hence, by Theorem 3.2(4), there exists a series ∑ an Pn with (an )n≥1 ∈ ∩ p>1 ℓ p and Pn (x, y) = Py (x − ξn ), so that a sequence of partial sums (Sνn )n≥1 converges uniformly on compact subsets of H to u. The proof of the statement for the derivatives follows from the fact that the sequence of harmonic functions (Sνn )n≥1 , being convergent to u uniformly on compact subsets of H , also satisfies that, for any multi-index α, ∂ α Sνn → ∂ α u in the same sense. The proof is complete.  6.2. The case of the ball Let us replace H in (6.1) by B, the unit ball of Rd , and ∂ H by S = ∂ B, the unit sphere in Rd , where now f is a continuous function on S. We recall that the function ∫ 1 1 − |x|2 u(x) = f (ζ ) dσ (ζ ), x ∈ B, ωd S |ζ − x|d u(x) = f (x), x ∈ S,

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where σ is the area measure on S, is harmonic on B and continuous on B. It is therefore the solution of the boundary value problem. The expression P(x; ζ ) :=

1 1 − |x|2 , ωd |ζ − x|d

x ∈ B, ζ ∈ S,

in the integrand is the Poisson kernel for B. For ζ ∈ S, the function P(·; ζ ) is harmonic on  Rd \ {ζ } and, moreover, S P(x; ζ ) dσ (ζ ) = 1. We can then write the solution of (6.1) as ∫ u(x) = P[ f ](x) := f (ζ )P(x, ζ ) dσ (ζ ). (6.3) S

If, on the other hand, u is a continuous function on B and harmonic on B, then, by the uniqueness of the solution of (6.1), u(x) = P[u| S ](x), where u| S is the restriction of u on S. Although the integral representation in (6.3) is not a convolution of the boundary data with the Poisson kernel for the ball, the integral can be broken into a finite number of integrals over pieces of S and then approximated uniformly over compact subsets of B by Riemann sums, where the ζ belong to a countable dense subset of S. Hence a theorem like Theorem 6.1 can be proved. Alternatively we may work as follows. Writing each point x ∈ Rd in spherical coordinates x = x(r, θ ) with θ = (θ1 , . . . , θd−1 ), where 0 ≤ θ j ≤ π for j = 1, 2, . . . , d − 2 and 0 ≤ θd−1 ≤ 2π , we have the following. d−2 Theorem 6.2. Let {sn }∞ n=1 be an enumeration of all rational vectors in [0, π] ∑ × [0, 2π ] and let ζn = ζ (1, sn ), n ∈ N. Let Pn (x) := P(x, ζn ). Then there exist series an Pn with (an )n≥1 ∈ ∩ p>1 ℓ p , having the property that, given a function u harmonic on B and continuous on B, there exists an increasing sequence (νn )n≥1 in N such that

∂xα

νn −

ak Pk (x) → ∂xα u(x),

(6.4)

as n → ∞

k=1

uniformly on compact sets of B for every multi-index α, |α| ≥ 0. Moreover, the set of such universal sequences (an )n≥1 is a dense G δ set in ∩ p>1 ℓ p , and it contains a dense vector subspace of ∩ p>1 ℓ p except the zero sequence. The proof is similar to that of Theorem 6.1, and is omitted. We conclude this section with a few comments. Remark 6.1. Although Theorem 6.2 is stated and proved in the context of boundary value problems for the steady-state equation, the result holds true in a more general setting. In particular, the series obtained is universal in the space of functions harmonic in the unit ball B. Indeed, every such function is approximated in the topology of uniform convergence on compact sets of B by functions harmonic on B and continuous on B. The same result also holds for harmonic functions on H . The analogous statement is also true for solutions of the heat equation in Rd × R+ . ∑ Remark 6.2. We note, as in Remark 3.1, that no universal series an Pn exists with ℓ1 coefficients. The supposition that it does leads to ∞ − n=1

|an Pn (x, y)| =

2 ωd+1

∞ − n=1

|an |

(|x − ξn

|2

y + y 2 )(d+1)/2

V. Nestoridis et al. / Journal of Approximation Theory 163 (2011) 1783–1797

≤

2

∞ −

ωd+1

n=1

= ay

−d

|an |

1797

1 yd

,

for the upper half space H , and ∞ −

an Pn (0) =

n=1

∞ 1 − an = b, ωd n=1

∑ for the ball, where a, b ∈ R, via the absolute convergence of the series ∞ n=1 an . But this contradicts that the partial sums are dense in the relevant spaces of harmonic functions. Note. An earlier version of this work was completed while the first and the last author were visiting the Department of Mathematics and Statistics of the University of Cyprus. It appeared as technical report: V. Nestoridis and V. Stefanopoulos, Universal Series and Approximate Identities, TR-28-2007, Department of Mathematics and Statistics, University of Cyprus. When some authors refer to “V. Nestoridis and V. Stefanopoulos, Universal Series and Approximate Identities”, they mean the above-mentioned technical report. In particular, “A.G. Bacharoglou, Approximation of Probability Distributions by Convex Mixtures of Gaussian Measures, Proc. AMS, 138, (2010), no. 7, 2619–2628” uses a lemma which does not exist in the present version. Acknowledgments V.N. and V.S. are grateful for hospitality at the University of Cyprus. We also thank K. Fokianos and A.M. Olevskii for several fruitful discussions we had while preparing this paper. V.N. and V.S. were partially supported by Caratheodory program #C164. S.S. was supported by the FWF Grant SISE S10602. References [1] D.L. Alspach, H.W. Sorenson, Nonlinear Bayesian estimation using Gaussian sum approximations, IEEE Trans. Automat. Control AC-17 (4) (1972) 439–448. [2] F. Bayart, K.-G. Grosse-Erdmann, V. Nestoridis, C. Papadimitropoulos, Abstract theory of universal series and applications, Proc. Lond. Math. Soc. (3) 96 (2) (2008) 417–463. [3] H.G. Feichtinger, A characterization of Wiener’s algebra on locally compact groups, Arch. Math. 29 (1977) 136–140. [4] H.G. Feichtinger, F. Weisz, The Segal algebra S0 (R d ) and norm summability of Fourier series and Fourier transforms, Monatsh. Math. 148 (2006) 333–349. [5] K. Gr¨ochenig, Foundations of Time–Frequency Analysis, Birkh¨auser, Boston, 2001. [6] K-G. Grosse-Erdmann, Universal families and hypercyclic operators, Bull. Amer. Math. Soc. 36 (3) (1999) 345–381. [7] J.P. Kahane, Baire’s category theorem and trigonometric series, J. Anal. Math. 80 (2000) 143–182. [8] J.T.H. Lo, Finite dimensional sensor orbits and optimal nonlinear filtering, IEEE Trans. Inform. Theory IT-18 (5) (1972) 583–589. [9] V. Nestoridis, C. Papadimitropoulos, Abstract theory of universal series and an application to Dirichlet series, C. R. Acad. Sci. Paris 341 (9) (2005) 539–543. [10] H. Reiter, J.D. Stegeman, Classical Harmonic Analysis and Locally Compact Groups, Oxford Science Publications, Clarendon Press, 2000.